# dependence on initial conditions of solutions of ordinary differential equations

Let $E\subset W$ where $W$ is a normed vector space^{}, $f\in {C}^{1}(E)$ is a continuous^{} differentiable map $f:E\to W$. Furthermore consider the ordinary differential equation^{}

$$\dot{x}=f(x)$$ |

with the initial condition^{}

$x(0)={x}_{0}$.

Let $x(t)$ be the solution of the above initial value problem defined as

$$x:I\to E$$ |

where $I=[-a,a]$. Then there exist $\delta >0$ such that for all ${y}_{0}\in {N}_{\delta}({x}_{0})$(${y}_{\mathrm{0}}$ in the $\delta $ neighborhood of ${x}_{\mathrm{0}}$) has a unique solution $y(t)$ to the initial value problem above except for the initial value changed to $x(0)={y}_{0}$. In addition $y(t)$ is twice continouously differentialble function of $t$ over the interval $I$.

Title | dependence on initial conditions of solutions of ordinary differential equations |
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Canonical name | DependenceOnInitialConditionsOfSolutionsOfOrdinaryDifferentialEquations |

Date of creation | 2013-03-22 13:37:19 |

Last modified on | 2013-03-22 13:37:19 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 7 |

Author | Daume (40) |

Entry type | Theorem |

Classification | msc 35-00 |

Classification | msc 34-00 |